Preheat the oven to 350°F with the oven rack in the center. Line three baking sheets with parchment paper and set aside.
Remove two of the refrigerated dough squares from the refrigerator at a time. Lightly flour a work surface and gently roll out one of the dough squares into a rectangle about 1/16th-inch thick and about with dimensions of about 12-by-15-inches (this doesn’t have to be perfect, and may be slightly different in size if your four pieces of dough weren’t divided evenly). Do your best to create straight edges and corners for your rectangle, and an even thickness, dusting with minimal flour as necessary. Arrange the rectangle so the longest edge is facing you with the shorter edges on either side.
Add ¼ of the filling to the rectangle and use your hands to spread it over the surface, leaving about a 1-inch border at the furthest edge clear of filling, but otherwise spreading the filling from edge to edge. Use the palms of your hands to firmly press the filling into the dough. This will make it easier to roll without the filling falling out too much.
Beginning with the edge closest to you, start rolling the dough tightly. Sometimes it’s easier to simply make a small fold across the length, using your dough scraper for assistance, and then continue with the rolling. When finished rolling, gently press the top of the roll to help seal.
Carefully move your roll to a clean piece of parchment or wax paper on your work surface. Lay it diagonally if you must so it doesn’t hang off the edges. Use your rolling pin to gently flatten the roll starting at the center and moving outward, and then again starting at the center and moving outward in the opposite direction. You are not rolling it out, but just using the natural momentum of the rolling pin to flatten it so it’s not rounded like a jelly roll.
Brush the top and sides of the roll with the beaten egg. Use a serrated vegetable cutter (if you have one–and if not, a dough scraper will work but won’t yield the pretty serrated edges) to cut the dough into 1 ¼-inch slices. Very gently use a small spatula or your dough scraper to remove each piece to the prepared pans. The end pieces aren’t pretty, but will still be delicious for the chef, so make sure you add those to your pan as well.
When finished with the first roll, dump out any extra crumbs from your parchment paper (as long as they’re not eggy) back into the bowl of filling, and use your dough scraper to clean any extra crumbs from you original work surface and add those back to the bowl as well.
Repeat the process with the remaining dough squares on your clean, lightly floured work surface (you can fold the parchment in half lengthwise and reuse it again once more, and then start fresh on dough #3 with a clean piece).